Covariance (2 pages)

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Covariance



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Covariance

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IV. Covariance V. Conditional Expectation VI. Normal Distribution


Lecture number:
3
Pages:
2
Type:
Lecture Note
School:
Cornell University
Course:
Econ 3120 - Applied Econometrics
Edition:
1

Unformatted text preview:

Econ 3120 1st Edition Lecture 3 Outline of Last Lecture I Marginal Distributions II Expectations and Variance III Covariance Outline of Current Lecture IV Covariance V Conditional Expectation VI Normal Distribution Current Lecture 2 3 Covariance When we are dealing with more than one random variable it is useful to summarize how these two random variables move together Suppose we have two random variables X and Y and we define E X x and E Y y The covariance between these two random variables is defined as Cov X Y xy E X x Y y We can also write the covariance as Cov X Y E X x Y E X Y y E XY x y 2 3 1 Properties of the covariance 1 If X and Y are independent then Cov X Y 0 This follows since E XY E X E Y when X and Y are independent 2 For any constants a1 b1 a2 b2 Cov a1X b1 a2Y b2 a1a2Cov X Y Now that we know about the Covariance we can define a third property of the variance 3 Var aX bY a 2Var X b 2Var Y 2abCov X Y One issue with covariance is that the units are difficult to interpret It turns out that we can scale covariance by the standard deviations of both variables and to get the unit friendly correlation Corr X Y xy Cov X Y sd X sd Y xy x y 2 4 Conditional Expectation In econometrics we often want to know how much one variable X tells us about another variable Y One way to do this is by using covariance and correlation but another concept we will be using a lot is conditional expectation Conditional expectation written as E Y X x often shortened to either E Y X or E Y x tells us the mean of Y conditional on some value of X The conditional expectation is defined in a similar way to the unconditional expectation above but using the conditional probability distribution functions discrete r v E Y x y y f y x continuous r v E Y x y y f y x Example Suppose we are studying the relationship between schooling and earnings and that hourly wages and schooling are our random variables How does the mean wage vary with the schooling level Our CEF conditional expectation



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